(ii) Determine the smallest value of n such that
a(n+1) &endash; a(n) is greater than 5000.

c) In the expansion of ,
state:

i) the number of uncollected terms
ii) the coefficient K in the collected term .

d) Determine the number of collected terms in the
expansion of .

2.

Georgania Higgenbottom served a variety of cool drinks at
her spring garden party. Each of the 120 guests selected
exactly one beverage, chosen from the following list:
ginsing go-getter, spinach surprise, cabbage combo, tomato
torment, and cucumber delight.

Georgania's head waiter, Christof, kept a tally sheet of
drink selections for the 120 guests.

a) Suppose Christof noticed that each of the beverages
had been selected by at least one guest. How many different
tally sheets could Christof have generated for the party's
drink selection under these circumstances?

b) Suppose instead that when Georgania screamed to
Christof, "What are our minimums on drinks?" he replied, "I
see that at least 10 people chose the ginsing go-getter, 20
or more went for the spinach surprise, more than 6 people
tried the cabbage combo, no one selected tomato torment, and
at least 15 drank cucumber delight." Under these conditions,
how many different tally sheets could Christof have
generated for the party's drink selection?

c) Try to picture Amiele, the beverage host, walking
around the table asking guests for their beverage choice.
How many guests would Amiele have to ask before he would be
assured that a beverage choice repeated itself? What
assumptions underlie your response?

3.

In the Ancient yet little-known game of gnikoj,
scoring occurs several ways. A player can earn 1 point for a
kaboom, 1 point for a laboom, and 1 point for
a maboom. A player earns 2 points for executing a
bifter or for executing a cifter. In
traditional gnikoj, there is no way to earn 3 points
on a single move, and the only way to earn 4 points is by
the rarely seen move called a gemserp. Points are
added and the player with the highest point total wins.

a) Show at least three different scoring sequences a
gnikoj player could execute and earn 11 points. Note
that the order in which points are earned is significant. A
kaboom followed by a cifter is a different
scoring sequence than a cifter followed by a
kaboom.

b) In gnikoj, play continues until an
essapmi is reached. Individual player totals could
exceed 100 points. Create a recursion relationship
T(n), including any required initial conditions, to
describe the number of different sequences of gnikoj
moves that result in a total of n points. Show
justification for your result.

4.

What is wrong with the following problem situation?

A survey of 144 new teachers in a
metropolitan school district determined that 16 new
teachers had been to Europe, 13 had been to Asia, and 17
had been to Africa. Of those who had been to Europe, 10
had been to Africa. Of those who had been to Asia, 9 had
been to Africa. Of all the new teachers surveyed, 5 had
been to Africa, Asia, and Europe, and 111 new teachers
had been to none of those continents.

Describe specifically and clearly what is wrong
and how you determined that.

5.

a) Using an unlimited supply of letters from the set
{A,B,C,D}, how many 7-letter sets are possible to create
with at least one C in the set?

b) Using an unlimited supply of letters from the set
{A,B,C}, how many 5&endash;letter words (meaningless or not)
are possible to create with exactly one A in the word?

6.

In a local chess tournament, two players compete until
one player wins 4 matches. With no regard for the ability of
the players, what fraction of all possible two-player
competitions will require exactly 7 matches?